Can colleges "see" ur SAT score before u do?

I earned a 1340 on the PSAT, and since then I have been receiving spam emails from colleges that search for students in that SAT score range (Barnard College, Boston University, University of Miami, Northeastern University, etc.).
But after completing the real SAT on April 5th, I have been receiving emails from colleges that search for students with much higher SAT scores. For example, I received such emails from Princeton University and University of Chicago, and the bottom of the emails state: “We received your contact information from the College Board’s Student Search Service.”
My question is: if these universities have JUST put me on their mass email list, is it possible it is because my April SAT score is higher (1400-1600), even though this score hasn’t been released to students?

No. It’s from the PSAT.

It is from your PSAT registration that you check the box to receive college information. Colleges do purchase mail list information with certain criteria. They can choose certain geographic area, score range, etc. They do not know your exact score unless you choose to submit to them.